A mathematical model to predict glioblastoma invasion : use of patient-specific diffusion tensor imaging data in simulations

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SCUOLA DIINGEGNERIAINDUSTRIALE E DELL’INFORMAZIONE CORSO DILAUREA MAGISTRALE ININGEGNERIAMATEMATICA

A Mathematical Model to Predict

Glioblastoma Invasion: use of patient-specific

diffusion tensor imaging data in simulations

Master thesis

RELATORE

Prof. Davide AMBROSI

CORRELATORE

Prof. Pasquale CIARLETTA

Dott.ssa Chiara GIVERSO

TESI DI LAUREA DI: Maria Cristina COLOMBO

MATR. 782447

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Alla mamma e al pap`a, a Fra e ad Emma, a Dolly

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List of Figures

1.1 How cancer starts . . . 12

1.2 How cancer spreads . . . 12

1.3 Glial tissue’s supportive functions . . . 15

1.4 Example of butterfly glioblastoma . . . 19

1.5 Glioblastoma resection . . . 21

2.1 Magnetic field gradients and their effects on the MRI signal . . . . 25

2.2 The relationship between water motion and gradient application . 26 2.3 Different degrees of DWI in relation to the b factor . . . 28

2.4 Diffusion tensor components . . . 30

2.5 Diffusion components in relation to fibers orientation . . . 31

2.6 Scheme of diffusion properties via ellipsoid visualization . . . 33

2.7 Depiction of the eigenvalues, mean diffusivity, FA and a color map of the fiber orientation . . . 33

3.1 Schematic diagram of cells’ behavior in relation to Σ . . . 45

3.2 Nutrients diffusion through ECM . . . 47

3.3 Exerted pressure in a glioblastoma . . . 50

4.1 T1 image used for mesh creation . . . 54

4.2 Segmented images of the brain. . . 54

4.3 Segmented glioblastoma tumor overlapped to the brain map . . . . 54

4.4 Mesh surface before and after smoothing and cleaning steps . . . 55

4.5 3D mesh before and after the refinement of the tumoral area . . . 56

4.6 Labeled mesh . . . 57

4.7 Diffusion tensor - medical images . . . 59

4.8 Patient-specific components of tensor D . . . 62

4.9 Mean Diffusivity . . . 62

4.10 Patient-specific components of tensor T . . . 63

4.11 Initial normal profile of the cellular fraction . . . 65

4.12 Initial distribution of φ on the brain mesh - plane xy . . . 65

4.13 Initial tumor shape . . . 66

4.14 Initial distribution of n on the brain - plane xy . . . 67

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5.1 Initial normal profile of φ . . . 75

5.2 Initial φ concentration on the brain mesh - plane xy . . . 75

5.3 Legend of the symbols ∆x, ∆y and ∆z . . . 76

5.4 How profiles are obtained . . . 81

6.1 Anatomy of the corpus callosum . . . 87

6.2 Frontal and occipital butterfly glioblastoma . . . 87

6.3 Fractional anisotropy - medical image . . . 88

6.4 Location of initial tumor . . . 89

6.5 Initial configuration of φ and n . . . 89

6.6 Plot of knover a clipped mesh along xy and yz planes. . . 91

6.7 φ concentration in relation to Txxvalues . . . 93

6.8 φ concentration in relation to Tiivalues . . . 94

6.9 φ concentration in relation to kn . . . 95

6.10 φ concentration at t=25 days in relation to kn- isotropic simulation 96 6.11 Manually segmentation of glioblastoma . . . 97

6.12 Different regions on the mesh . . . 98

6.13 3D sketched of the tumor mass and survivor cells . . . 98

6.14 knconcentration over a clipped mesh . . . 99

6.15 Snconcentration over a clipped mesh . . . 100

6.16 Diagonal components of tensor T . . . 100

6.17 Diagonal components of tensor D . . . 101

6.18 Refined mesh - Resection test . . . 101

6.19 φ and n concentration development in plane xz - resection test . . 104

6.20 φ and n concentration development in plane yz . . . 104

6.21 Detail of φ and n at t=20 days . . . 105

6.22 Clipped thresholded φ and n concentration in relation to Dxx . . . 105

6.23 Clipped thresholded φ and n concentration in relation to kn . . . . 106

6.24 Volumetric growth of glioblastoma . . . 106

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List of Tables

1.1 Type of cells, tissues and tumors . . . 14

1.2 WHO grading system for gliomas . . . 16

1.3 Incidence rates, age and sex, and survival of patients with gliomas 17 3.1 Parameters Estimation . . . 49

5.1 Biological parameters - Sensitivity analysis I . . . 77

5.2 Biological parameters - Sensitivity analysis II . . . 77

5.3 Numerical results concerning φ - Sensitivity analysis I . . . 78

5.4 Numerical results concerning φ - Sensitivity analysis II . . . 79

5.5 φ profile - Sensitivity analysis I . . . 82

5.6 Plots of n variables at t = 0 - Sensitivity analysis II . . . 83

5.7 φ and n profile at the final time step - Sensitivity analysis II . . . 84

6.1 Diagonal components tensor T in the ROI . . . 90

6.2 φ and n concentration, clipped mesh xy plane - butterfly and isotropic simulations . . . 92

6.3 Tumor volume overlapped to the maps of Tii. . . 94

6.4 φ and n concentration at different time steps - Resection test . . . 103

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Even though brain tumors account for only 2-3% of all cancers, they are re-sponsible for 7% of the years of life lost from cancer before the age of 70. Among them, the most aggressive is the glioblastoma, a highly malignant cancer that arises in the neuroglia, the supportive tissue of the neurons. Glioblastoma presents long extensions that infiltrate deeply the white matter, following the alignment of the fibers. From the medical viewpoint, this peculiarity makes it difficult to treat. For the same reasons, in the last years, biomathematical modeling applied to infiltrative brain tumor has gained in importance. Indeed, a good model could offer a better understanding of the microstrucutral dynamics of the cancer and thus it could be helpful to predict its evolution. In this study, we propose a diffuse interface binary mixture model which consists of a fourth order non-linear equation for the can-cerous cellular fraction coupled with a reaction diffusion equation for the nutrient component. The model takes into account the mechanical dynamics, e.g. adhe-sive forces or viscous interactions among cells, and the chemotactic cellular move-ment in response to certain environmove-ment factors. Moreover, we include brain tissue heterogeneity and anisotropy in the model by the introduction of patient-specific diffusion tensor imaging data, thanks to which we manage to probe brain fibers architecture. The aim of this research is to demonstrate the importance of consid-ering anisotropy, heterogeneity and patient-specific data into mathematical models in order to better predict the tumor growth. Specifically, we deal with the theoret-ical and the numertheoret-ical framework of the mathemattheoret-ical model proposed. Starting from a real MR of a patient affected by glioblastoma and using imaging techniques, we create a patient-specific computational mesh and we extract the necessary data from the DTI medical images. Then we develop numerical codes making use of an open-source software name FEniCS. To study the anisotropic development of the tumor in relation to the biological parameters presented in the model, we per-form a sensitivity analysis on a homogeneous geometry. Finally, we provide two numerical tests that simulate common clinical situations.

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Sebbene i tumori cerebrali rappresentino solamente il 2-3% delle diagnosi tu-morali, essi sono i responsabili del 7% di morti annue causate dal cancro prima dei 70 anni di età. Tra di essi il più aggressivo è il glioblastoma, un cancro altamente maligno che si sviluppa nella neuroglia, il tessuto di supporto dei neuroni. Il glio-blastoma presenta lunghe estensioni che infiltrano la materia bianca in profondità, seguendone lallineamento delle fibre, e che rendono questo tipo di tumore molto difficile da trattare. Per questa ragione, negli ultimi anni, la modellizzazione ma-tematica dei tumori cerebrali infiltrativi ha acquisito sempre più importanza. Un buon modello può infatti offrire una migliore comprensione delle dinamiche mi-crosttrutturali del cancro e di conseguenza potrebbe essere utile nel predire la sua evoluzione. In questo progetto, proponiamo un modello di miscela binaria ad inter-faccia diffusa che consiste di unequazione del quartordine per la frazione volumica cellulare accoppiato con unequazione di diffusione reazione per la componente dei nutrienti. Il modello prende in considerazione le dinamiche meccaniche, come le forze di adesione o le interazioni viscose che hanno luogo tra le cellule, e il movimento chemotattico cellulare causato da fattori chimici presenti nellambiente extracellulare. Inoltre, abbiamo introdotto nel modello leterogeneità e lanisotro-pia, caratteristiche peculiari del tessuto cerebrale, utilizzando i dati medici reali del tensore di diffusione, grazie al quale possiamo anche conoscere larchitettura delle fibre cerebrali. Lo scopo di questa ricerca è dimostrare limportanza di introdurre leterogeneità, lanisotropia e i dati paziente-specifici nel modello matematico cos`ı che la predizione della crescita tumorale sia migliore e più veritiera. Nello specifi-co, in questo lavoro di tesi, ci siamo occupati dello sviluppo teorico e numerico del modello proposto. In particolare, attraverso tecniche di imaging medico, abbiamo creato una mesh computazionale di un cervello estrapolando la geometria da una risonanza magnetica di un paziente affetto da glioblastoma e abbiamo estratto i dati reali da introdurre nel modello dalle immagini mediche del tensore di diffusione. Abbiamo inoltre sviluppato codici numerici utilizzando un software open-source chiamato FEniCS. Per studiare la crescita anisotropa del tumore in relazione ai parametri biologici presenti nel modello, abbiamo realizzato unanalisi di sensiti-vità su una geometria omogenea. Infine, abbiamo proposto due test numerici che simulano situazioni cliniche comuni.

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Acknowledgments

Desidero ringraziare tutti coloro che mi hanno sostenuto e aiutato durante questo periodo di tesi.

Innanzitutto, vorrei ringraziare il mio relatore ufficioso, il professore Pasquale Cia-rletta, per avermi proposto un argomento cos`ı interessante e ostico. Le nozioni matematiche e numeriche che ho acquisito in questo anno sono notevoli, ma ancor più notevole è stata l’esperienza da ”ricercatrice” che ho sperimentato. Ho potuto comprende, almeno un po’, cosa significa essere un ingegnere matematico che fa ricerca, che tenta di comunicare con altre figure che molto spesso non capiscono il suo linguaggio, che cerca di trovare modelli là dove nessuno ci ha mai pensato prima. Ho avuto modo di mettermi a confronto e di conoscere il mondo della medicina, che sempre mi ha affascinato e spaventato.

Colgo l’occasione dunque per ringraziare i medici dell’Istituto Neurologico Carlo Besta per essere stati disponibili e per aver soddisfatto le mie richieste. In partico-lare, ringrazio il dottor Carlo Boffano per aver fornito i dati medici di cui necessi-tavo e senza i quali questo progetto non avrebbe potuto essere realizzato; e il dottor Francesco Acerbi per aver risposto ai miei dubbi e per aver dimostrato particolare interesse e fiducia verso questa ricerca.

Vorrei ringraziare ora le persone che mi hanno aiutato nell’affrontare e nello svilup-pare questo lavoro. Ringrazio Elena Faggiano per avermi insegnato le tecniche base di imaging e per essere stata disponibile ad incontrarmi, e Matteo Manica, il quale indirettamente mi ha fornito i codici numerici dai quali sono partita. Pi`u di tutti vorrei ringraziare la mia correlatrice, la dottoressa Chiara Giverso, il cui aiuto

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è stato fondamentale per la buona riuscita di questo progetto. La ringrazio pro-fondamente perché ha sempre risposto ad ogni mia domanda e ha sempre sanato ogni mio dubbio, la ringrazio perché mi sono sentita libera (forse fin troppo) di es-primere ogni mia perplessità e paura, la ringrazio perché ha ascoltato i miei sfoghi e mi ha salvato da possibili esaurimenti nervosi. A lei in parte devo il fatto che sono arrivata fin qui.

Desidero ora ringraziare i miei cari e i miei amici, che insieme a me hanno vissuto e affrontato il periodo universitario.

Un grazie enorme va alla mia famiglia, perché, nonostante tutti i difetti che abbi-amo, siamo la famiglia migliore del mondo. Vorrei ringraziare il papà per avermi sempre sostenuto, economicamente e moralmente, e per avermi dato la possibilità di arrivare fino a qui, fino ai 25 anni, senza fare nulla e senza farmi mancare nulla. Ringrazio la mamma per essere la mamma, per essere ”tutto”, per avermi aiutato sempre, per aver dato importanza ad ogni mio respiro e ad ogni mia lacrima, per aver vissuto con me ogni mio affanno e per avermi spronato a fare meglio quando avevo paura. La ringrazio perché mi diceva sempre di smettere di studiare e perché mi trascinava al supermercato per farmi prendere un po’ d’aria. Ringrazio mia sorella maggiore Francesca, per aver condiviso con me tutta la sua vita, per essere unica, per avermi dato consigli, per aver instaurato con me un rapporto speciale, ir-riproducibile, vero. Ringrazio mia sorella minore Emma, perché è il sole di questa casa, perché riesce a farci ridere e a farci litigare, perché ci fa impazzire e ci fa sognare, perché mi fa vivere il liceo una seconda volta e perché, da quando Fra si è laureata, ha condiviso con me il tavolo della cucina. Ringrazio Dolly, anche se non c’è più. Mi manca da morire, ogni giorno di più. È il ricordo più vivido e ricor-rente nei miei pomeriggi di studio. Mi ha tenuto compagnia nei periodi invernali dormendo sulla sedia al mio fianco, quando la mia famiglia andava a fare compere il sabato pomeriggio, e si è fatta invidiare nei periodi estivi quando mi preparavo per le sessioni estive e lei sonnecchiava nell’angolo più fresco della casa.

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Desidero ringraziare i miei compagni e amici di università e in particolare il gruppo degli ”Ingegneri Matematici Belli Ma Non Troppo” per aver reso questi anni di studio un po’ meno sofferti, trasformando le sere di studio in biblioteca in archi-party. Vi ringrazio perché abbiamo saputo mantenere i nostri obblighi senza dimenticare di divertirci. Vi ringrazio perché avevate un modo di vivere e di pensare uguale al mio, ed è stato bello trovare un gruppo di amici cos`ı affiatato in un periodo della vita dove le amicizie vanno disperdendosi. Vi ringrazio per tutto, per avermi dato un letto a Milano, per aver reso questa città anche un po’ mia, per le serate all’Alcatraz, per le cene di Natale, per le cene e basta, per le gite fuori porta, per ogni esame affrontato, per ogni minima stupidata che abbiamo fatto, per tutto. Vorrei ringraziare il mio gruppo di calcolo ed, in particolare, il mio gruppo del Compatto per aver condiviso le fatiche del quarto anno e per aver odiato e continuare ad odiare quelli di finanza. Vorrei ringraziare sopra tutti e tutto Elisa e Francesca, perché siete ”belle belle belle”. Ringrazio Elisa perché è stata un appoggio sicuro in questi ultimi anni, e, nonostante all’inizio ci tenevamo alla larga forse perché io piangevo troppo e lei aveva collanine con croci al contrario, si è rivelata un’amica sincera. E’ la mia Miley. Ringrazio Francesca perché mi fa sentire importante e perché parla tanto. Non mi sono mai annoiata con lei e mi fa vedere la vita da un punto di vista diverso. Mi ha spiegato che differenza ci fosse tra cipria e fard e mi ha sgridato quando avevo comportamenti infantili.. anche se ci resto male ogni volta, so che lo fa per farmi diventare migliore. Ringrazio Luigi, per essermi stato vicino in questi ultimi due anni durante i quali ha saputo sempre ascoltare le mie lamentele e i miei deliri. Ringrazio la sua sensibilità e la sua ca-pacità di analizzare la vita e le persone che ci circondano.

Ringrazio le mie amiche del liceo, Silvia, Dani e Lara, per esserci sempre e per essere cresciute con me in questi 10 anni. Più di tutti, un grazie sincero va a Silvia, perché la sua amicizia è stata unica e preziosa.

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In questi anni di frequentazione, ho avuto modo di staccare la spina dal mondo astratto della matematica e di imparare a muovermi nella geometria reale, di avere coscienza dello spazio, impegnandomi in qualcosa di fisico e accorciando le dis-tanze tra pensiero e movimento.

Ed infine ringrazio Jacopo, solo perché vuole essere ringraziato per battere il suo stesso record di ringraziamenti su tesi universitarie. Dovrei anche ringraziare il suo computer, perché grazie ai suoi quattro core, mi ha aiutato moltissimo ad abbattere i tempi delle simulazioni. Dovrei forse però ringraziare Jacopo anche per essermi stato sempre vicino quest’anno e per avermi sempre e incondizionatamente sopportato, per aver cercato con me soluzioni che da sola non riuscivo a trovare, per aver accolto le mie lacrime sulla sua spalla e per aver risposto ad ogni mio sorriso con un sorriso. Lo ringrazio perché quando ho perso le speranze e quando sono stata stanca di lottare e di pensare, la sua presenza mi ha fatto dimenticare tutto e tutti. Lo ringrazio perché è curioso ed intelligente e ha sempre dimostrato interesse alle cose che facevo e studiavo, nonostante non ci capisca molto di matematica. Lo ringrazio perché è un cuc, e spero che lo rimanga per sempre.

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Introduction

Primary brain tumors are the most aggressive and lethal forms of cancer [75], and, although they account for only 2-3% of all human malignancies, they are nev-ertheless responsible for 7% of lives lost from cancer before the age of 70 per year [19]. In human brain, moreover, more than 120 different tumor types can be found among which gliomas are the most prevalent form [23]. They are tumors that arise from neoplastic glial cells, the supportive tissue of the neurons, and account for the 30-40% of all brain and central nervous system primary tumors and the 80% of the all malignant brain tumors [21, 23].

The most common and also the most malignant glioma is the glioblastoma, which will be the clinical focus of this dissertation. Despite of therapies (surgery resec-tion, chemotherapy or radiotherapy), glioblastoma has a life expectancy, after been diagnosed, of only fourteen months [69, 75], while only 5% of patients has a five years survival [21]. Thus, this disease represents a real challenge for present day oncology. Moreover, its capability of penetrating diffusely along the white mat-ter fibers leads to a wide tumoral area underestimated by conventional imaging technology, such as computed tomography (CT) and magnetic resonance imag-ing (MRI). Consequently, invasive glial neoplasms are difficult to be completely resected, and while all visible tumor cells can be removed, the invisible infiltrat-ing tumor components are left behind. For this reason, the extent of resection in glioblastoma medical treatment has been a big dilemma in the last decades. Fur-thermore, another challenging aspect of surgery is related to the location of the tumor. As a matter of fact, tumor resection could be potentially harmful if the

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mor is located near a sensitive area of the brain such those which control language, movement, vision or other important functions.

To better detect tumors and evaluate their invasiveness, in the early 90s a new imag-ing technique called Diffusion Tensor Imagimag-ing (DTI) was developed [30]. The DTI is the only non-invasive method for characterizing the microstructural organization of tissue in vivo, so it is understandable why this technique has attracted huge in-terest and has enjoyed a rapid uptake by clinical and neuroscientific communities.

At the same time, a big help to present day oncology is potentially represented by biomathematical modeling. During the last decades, indeed, the capability of tumor to grow and invade the surrounding tissue has gained also the attention of the mathematical and the physical research communities. The goal of this new field of research is to understand the dynamics behind tumor growth and expansion in relation to the specific features of the environment in which the neoplasm is evolv-ing in order to provide a good mathematical model able to predict tumor evolution, to improve therapy planning and to prognosticate the result of a specific form of treatment. Thus, during the last fifty years, driven by the ambition of providing a good and truthful model, theoreticians have proposed different mathematical mod-els, covering various morphological and functional aspects of carcinogenesis. As concerns the field of infiltrative brain tumor, such as gliomas, the basic spatio-temporal model was suggested by J.D. Murray in the early 1990s. He proposed a simple conservative diffusion equation for cells concentration observing that the main aspect of the cancer was the uncontrolled proliferation of cells and the capa-bility of them to invade and diffuse in the host tissue. The model can be stated as: the rate of change in tumor cell population =

the diffusion (motility) of tumor cells + the net proliferation of tumor cells. Under the assumption of a classical gradient-driven Fickian diffusion, this

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state-ment leads to the mathematical formulation of the following equation: ∂c

∂t = ∇ · (D∇c) + ρc (1)

where c(x, t) is the tumor cell density at location x and time t, D is the diffusion coefficient representing the net motility of glioma cells and ρ represents the net pro-liferation rate of glioma cells [22]. In the last ten years, Swanson et al. have been working on the development of this kind of model, introducing brain anisotropy [71], angiogenesis [73], and modeling therapeutic response [72].

In contrast to diffuse models, in the mid-1990s, multiphase models, whose the-oretical framework has to be found in the theory of mixture, started to be used in biological research and in particular in the study of tumors. The multiphase ap-proach was novel insofar as the tumor is considered as a continuum comprising two or more interacting constituents, called phases. The mechanical aspects of bi-ological tissues are also introduced in governing equations, which are now driven from mass and momentum balances. Among multiphase models, an interesting model was proposed by Byrne and Preziosi [8] in the early 2000s. They proposed a binary model in which the tumor cells phase and the extracellular water phase are taken into account and they showed that cellular motion is a consequence of interaction between cells instead of being determined by random motility. In the same period Ambrosi and Preziosi [1] have explained the necessity to propose some postulates for cell velocity and the corresponding displacement field in order to close the model, derived from the mixture theory. In the same area of research, in 2007 Lowengrub et al. [83] developed, analyzed and numerically simulated a diffuse interface continuum model, which was derived using the energy variation statement and thus it was thermodynamically consistent.

Following [1, 8, 83] in the present work we propose, analyze and simulate a diffuse interface binary mixture model of glioblastoma tumor growth. The model is well posed and consists of a fourth order non-linear equation for the tumoral

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cells’ volume fractions coupled with a reaction-diffusion equation for the nutrient component. We consider the oxygen as the main source of supply and we hy-pothesize that the vasculature is homogenous in the whole domain. The model takes into account the mechanical aspects involved in tumor evolution, such as ad-hesive forces or viscous interactions, and the micro-local environment conditions such as the oxygen levels. Hence, in the governing equation of the cellular frac-tion we consider a term of flux of mass which, biologically, is due to a particular phenomenon of cellular migration called chemotaxis. Indeed, it has been experi-mentally demonstrated [16] that a cell can perform a directed migration in response to certain increasing gradient of chemoattractants, such as nutrients. Furthermore, observing that the glioblastoma is a highly infiltrative tumor, whose extensions dif-fuse deeply along white matter structures, we introduce the concept of anisotropy thanks to patient-specific diffusion tensor imaging data. As mentioned, this tech-nique provides a tensor that describes the spatial diffusion of the water molecules in cerebral tissues and, comparing the diffusion values along each direction, we are able to know which is the structure of the fibers and which are the preferential di-rections of movement. Indeed, a water particle will move more easily along a fiber than perpendicularly to it because, along that path, it will not find any obstacles. In consequence, handling with the diffusion tensor we obtain a new tensor that we call tensor of preferential directions and that we include in the chemotactic term. By the inclusion of the anisotropic tensor of preferential directions, the model is thus able to reproduce the cellular motion along the fibers of the white matter. Finally, we suppose that the oxygen particles coherently with the water molecules and thus the diffusive behavior of the nutrients can be actually described by the water diffu-sion tensor, obtained directly from the medical images.

In our work we also deal with the implementation of the computational mesh and the numerical codes necessary to simulate the model. In particular, we obtain the computational mesh extracting the geometry from a magnetic resonance of a pa-tient affect by glioblastoma, gently provided by Istituto Neurologico Carlo Besta,

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and we discretize and numerically solve the problem using the finite elements method implemented in FEniCS. Consequently, we focus on simulating two spe-cific clinical cases: the first concerns the growth of a butterfly glioblastoma, a kind of astrocytoma that arises in the midline of the brain and assumes a peculiar shape; the second deals with a common medical situation in which the glioma is not com-pletely removed by surgical intervention and tumoral cells are left behind. We thus investigate the potential occurrence of metastasis in the site of resection.

The work is organized as follows: in Chapter 1, we describe the biological framework of the research, such as the phenomenon of carcinogenesis, and in par-ticular we focus on brain tumors and on glioblastoma malignancy and features. In Chapter 2, diffusion MRI technique is briefly described and the details on dif-fusion tensor imaging, useful to understand the mathematical model, are given. The first part of Chapter 3 is dedicated to the theory of mixture while in the sec-ond part the mathematical model is introduced. In the last part of this chapter the ranges of the biological parameters of the glioblastoma are discussed. In Chapter 4 the numerical formulation is provided. In the first part we describe the imaging and the computational techniques used to create the 3D brain mesh. In the second part of the chapter, it is explained how the diffusion tensor information are intro-duced in the model and how the tensor of the preferential directions is obtained. Finally in the last part of Chapter 4, the continuous and discrete weak formulation is developed and some details about the numerical method adopted are provided. In Chapter 5, the sensitivity analysis to the biological coefficients present in the model is presented. In Chapter 6 we describe and numerically simulate the cases of interest mentioned before. All the codes we make use of and more details about numerical methods are provided in Appendix A.

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Biological Framework

In this chapter we provide a biological framework of the problem in order to better understand which reasons are guiding our work and why a mathematical model of glioblastoma is highly required. So, in the first section we briefly explain the process of carcinogenesis, which literally means ”the creation of a cancer”. In the second section we focus on brain tumors and, among them, we concentrate on a specific group of brain cancers named gliomas. In the third section we illustrate the features of the glioblastoma, whose biological description is the object of this dissertation.

1.1 What is a cancer?

A multicellular organism can thrive only when all its cells function in accor-dance with the rules that govern cell growth and reproduction. In a healthy body cells control their proliferation and program their death (apoptosis) in the various tissues so as to optimize body repair and healing. Sometimes it can happen that this process breaks down, i.e. a mutation or an epimutation occurs in the DNA of a cell and it acquires abnormal functions. In a normal cell, when the DNA is dam-aged the cell either repairs the harm or the cell dies. In abnormal cells, the DNA is not repaired and the cell doesn’t die as it should but it survives and continues to

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proliferate, generating new cells that have the same damaged DNA. This leads to a mass of abnormal cells that grows out of control: the tumor (figure 1.1).

A tumor can be benign, not cancerous, which grows slowly and typically has clear borders that do not spread into other tissues, or malignant, i.e. the cancer, which grows rapidly, invades the surrounding tissue and metastasizes.

The formation of a cancer is a multistage process in which abnormal cells change the micro-environment to favor their survival. As a matter of fact, at the be-ginning, the cancer is just a microscopic nodule which does not have access to the vascular network and receives nutrients and growth factors via diffusion through the host (healthy) tissue. For example, the typical distance an oxygen molecule will diffuse before being uptake is approximately 100µm [13, 17, 57]. As time proceeds, tumor cells accumulate and the insufficiency of the existing vasculature to supply all cancerous cells may cause acute and chronic lack of oxygen (leading to hypoxia) and nutrient (e.g. glucose, leading to hypoglycemia). An avascular tumor cannot grow beyond a certain size, generally 1-2 mm3 [46]. Thus, a tumor only grows further if the cancerous cells acquire through mutations the ability to release pro-angiogenic growth factors (TAFs) in order to drive angiogenesis. The angiogenesis is the formation of new capillaries throughout the tumor mass which is induced by growth factors (e.g. VEGF) secreted by tumoral cells [80]. In this way the tumor has a direct supply of nutrients and growth promoting factors. Once a tumor is vascularized, it can grow larger and even shed cells into vessels, leading to satellite tumors in distant parts of the body (metastases) (figure 1.2). The formation of metastasis is the predominant cause of mortality due to cancer [40].

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Figure 1.1: Physiologically cells grow and multiply in an orderly way. If a mutation in the DNA of a cell occurs, this mechanism breaks down and cells grow out of control. A mutation in the DNA could be caused from exposure to exogenous agents such as UV light, X-rays, chemicals, tobacco products, and viruses or for endogenous factors (naturally occurring), figure from [50].

Figure 1.2: Once a tumor is vascularized, cancerous cells manage to move through-out the body using the blood or lymph systems, destroying healthy tissue in a pro-cess called invasion and it leads to metastasis, figure from [50].

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1.2 Introduction to brain tumors

1.2.1 An overview of brain tumors

There are over than 120 types of brain tumors and spinal cord tumors: they can appear in the brain itself or in lymphatic tissue, in cranial nerves, in the meninges or in the pituitary or in the pineal gland. Brain tumors are named after the type of cells they arise from or the area they grow in. For instance, gliomas, the majority of malignant brain tumors in adult, take the name from the neoplastic glial cells as well as meningiomas take their name from the meninges, thin layers that envelope the central nervous system (CNS) [79]. Tumors can be divided in primary tumors, i.e. tumors that start in cells of the brain and may spread to other parts of the brain or to the spine, but rarely to other organs, or metastatic or secondary brain tumors, that begin in another part of the body and then spread to the brain [66]. In table 1.1 we classificate and describe the principal primary CNS tumors on the basis of the type of cell or tissue they arise from [79].

Concerning the epidemiology of the disease, in Europe, the standardized incidence of primary CNS cancers ranges from 4.5 to 11.2 cases per 100,000 men and from 1.6 to 8.5 per 100,000 women, while the 5-year survival rate is 17% for males and 19% for females. The statistic estimations have been obtained by Crocetti et al. [14] studying patients affected by CNS cancers in the period between 1995 and 2002.

1.2.2 Classification of gliomas

Gliomas make up 80% of all malignant brain tumors [21] and, despite their frequency, the etiology of these tumors remains largely unknown. The common gliomas affecting the cerebral hemispheres of adults are named ”diffuse” gliomas due to their propensity to infiltrate extensively throughout the brain parenchymia. These gliomas are classified histologically as astrocytomas, oligodendrogliomas,

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Tumor Type Cell / Tissue

Origin Cell / Tissue Description Tumor Description Gliomas Neuroglia It is the supportive (gluey)

tissue of the brain (figure 1.3). It maintains the ionic milieu of nerve cells, it modulates the rate of nerve signal propagation and the synaptic action by control-ling the uptake of neuro-transmitters, and prevents from neural injury [61].

see subsection (1.2.2)

Medulloblastoma Neuroectodermal cells

They are very early forms of nervous system that are probably involved in brain cell development.

It develops in the cere-bellum and it is highly malignant. It affects children.

Meningioma Meninges They are layers of tissue that line and protect the brain.

It accounts for 1 out 3 primary brain tu-mors and they are usually benign. Many meningiomas are asymptomatic. Piutary tumors Piutary gland

and hypothala-mus

They both make hormones that help regulate the activ-ity of many other glands in the body, such as thyroid gland.

They account for the 8% of tumor and they are nearly benign. Symptoms are related with an excess or a lack of hormones produced by the glands.

Pineal tumors Pineal gland It is a small endocrine gland situated deep within the brain, between the cerebral hemispheres. It makes mela-tonin, the hormone that reg-ulates sleep.

They accounts for only 1 to 2% of brain tumors.

Schwannomas Schwann cells These cells surround and insulate cranial nerves and other nerves.

They make up about the 8% of CNS tu-mors. They are al-most always benign and can arise in any nerves. In relation to the func-tion of the nerve they start from, Schwanno-mas can cause differ-ent sympstoms, such as acoustic or balance problems.

Table 1.1: Classification and description of the principal primary Central Nervous System tumors on the basis of the type of cell or tissue they arise from [79].

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ependymomas, or tumors with morphological features of both astrocytes and oligo-dendrocytes, termed oligoastrocytomas. Astrocytomas arises from astrocytes, star-like cells (figure 1.3) that help support and nourish neurons and maintain an appro-priate chemical environment for neuronal signaling. Oligodendrogliomas are tu-mors formed by oligodendrocytes, the cells that make myelin, a fatty substance that surrounds and isolates the nerve cell axons and which helps neurons to send elec-tric signals. Ependymomas come from ependymocytes, the glial cells that make up the ependyma, the membrane that lines the ventricles of the brain and the central canal of the spinal cord. Gliomas are further categorized on the basis of the patho-logic evaluation and the histopatho-logical degrees of malignancy of the tumor. The most widely used current classification of human gliomas is the one proposed by the World Health Organization in 2007 [39] which we schematize in table 1.2. In table 1.3 we report population-based incidence rates (per 100000 person per year), age, sex and survival of patients which have been affected by gliomas in 1992-1997 in US [52] .

Figure 1.3: Neuroglia performs a supportive function for neurons. It is composed by microglial, that is the main form of active immune defense in the central nervous system and accounts for the 10-15% of the glial cells, and macroglial, derived from the ectodermal tissue, that is mainly composed by astrocytes (cells with a star-like appearance), oligodendrocytes, ependymocytes and Schwann cells. Figure taken from [61].

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WHO grading system

Grade Description Type of glioma

I Benign and slow growing

Cells look almost normal under a microscope Usually associated with long-term survival Rare in adults

Pilocytic astrocytoma

II Relatively slow growing

Sometimes spreads to nearby normal tissue and comes back (recurs)

Cells look slightly abnormal under a microscope Sometimes comes back as a higher grade tumor

Diffuse astrocytoma Oligodendroglioma Oligoastrocytoma

III Malignant = cancerous

Actively reproduces abnormal cells

Tumor spreads into nearby normal parts of the brain Cells look abnormal under a microscope

Tends to come back, often as a higher grade tumor

Anaplastic Astrocytoma Anaplastic Oligodendroglioma Anaplastic Oligoastrocytoma Anaplastic Ependymoma IV Most malignant, fast growing

Easily spreads into nearby normal parts of the brain Actively reproduces abnormal cells

Cells look very abnormal under a microscope Tumor forms new blood vessels to maintain rapid growth

Tumors have areas of dead cells in their center (called necrosis)

Glioblastoma - Gliosarcoma

- Giant Cell Glioblastoma

Table 1.2: WHO grading system applied to gliomas: grade I is assigned to the more circumscribed tumors with low proliferative potential, grade II defines diffusely infiltrative tumours, grade III is assigned to those showing anaplasia and mitotic activity and grade IV (glioblastoma) describes tumours that show microvascular proliferation and/or necrosis [39, 21].

1.3 Glioblastoma Multiforme

In the present work, we focus on glioblastoma (GBM) tumor, an astrocytoma grade IV, the most common and aggressive among gliomas, that accounts for 15% of all primary central nervous system tumors and 55% of all gliomas [21].

On the basis of clinical presentation, glioblastomas have been further subdivided into the primary or secondary GBM subtypes. The vast majority of glioblastomas are primary glioblastomas (90% of GBMs) which develop rapidly de novo in

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el-Tumor WHO grade Incidence Rate M:F ratio Mean age at diagnosis Survival 1 year 2 years 5 years 10 years Pilocytic astrocytoma I 0.23 1.09 17 95% 93% 89% 86% Diffuse Astrocytoma II 0.13 1.46 47 73% 60% 45% 34% Oligodendroglioma II 0.34 – 42 88% 80% 66% 47% Ependymoma/ Anaplastic Ependymoma II/ III 0.23 1.29 35 86% 79% 66% 55% Anaplastic Astrocytoma III 0.49 1.20 50 60% 43% 28% 19% Anaplastic Oligodendrocytes III 0.10 1.15 46 75% 57% 38% 25% Glioblastoma IV 2.96 1.26 62 28% 8.2% 2.9% 1.8%

Table 1.3: Population-based data of incidence rates (per 100,000 person per year), age and sex, and survival of patients with gliomas in 1992-1997 in United States adjusted to the 2000 US (see [52])

derly patients, without clinical or histologic evidence of a less malignant precursor lesion. In contrast, secondary glioblastomas progress from low-grade diffuse as-trocytomas or anaplastic asas-trocytomas. They manifest in younger patients, have a lesser degree of necrosis, are preferentially located in the frontal lobe, and carry a significantly better prognosis. Histologically, primary and secondary glioblas-tomas are largely indistinguishable, but they differ in their genetic and epigenetic profiles [53, 55].

Glioblastoma was observed for the first time in 1863-1865 by Virchow, known as the father of pathology, who was the first to identify a large group of intracra-nial tumors whose origin was in the glial tissue of the brain. He gave the name gliomato these new growths and he described such neoplasm as slow-growing, in-filtrating vascular masses with hemorrhages, cysts and degenerated areas. In 1925 Globus and Strauss [15] proposed the name spongioblastoma multiforme since, to the naked eye, glioblastoma appears as a well circumscribed mass, globular or spherical, with a highly variegate cut surface due to necrosis, fatty degeneration and hemorrhages and it is characterized by a necrotic core surrounded by

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anaplas-tic cells [27]. In 1926 the term glioblastoma multiforme was introduced as an alternative form of spongioblastoma by Percival Bailey and Harvey Cushing [2], two American surgeons who created the first major system of classification for brain tumors. They claimed that glioblastoma originates from primitive precursors of glial cells called glioblasts. In 1979 the first edition of the World Health Organi-zation (WHO) Classification of Tumors of the Nervous System was published and glioblastomas were not yet recognized as astrocytic neoplasms but listed in a group of ”poorly differentiated and embryonal tumors ” [55]. Only after the introduction of immunohistochemistry it has been discovered that glioblastoma is actually an extreme manifestation of anaplasia and dedifferentiation of astrocytes.

For unknown reasons, glioblastoma occurs commonly in Caucasians, Asians males over 50 years old, without any genetic predisposition [54]. The prevalent symptom is a progressive memory, personality, or neurological deficit [66], in addition to common symptoms such as seizure, nausea, vomiting and headache. Glioblastoma has median survival of about 14.6 months while for 30% of the cases survival could reach two years, [33]. However, there are reports that some patients can live for more than 10 years after diagnosis. A common feature among those patients is that they are typically younger than 40 years at the time of diagnosis [9, 65].

1.3.1 Hallmark features

Glioblastoma is defined by the hallmark features of uncontrolled cellular pro-liferation, diffuse infiltration, propensity for necrosis, robust angiogenesis, intense resistance for apoptosis and rampant genomic instability [19]. As reflected by the old epithet multiforme, GBM presents with significant heterogeneity on the cy-tophatologycal, transcriptional and genomic levels. The tumor may take on a vari-ety of appearances, depending on the amount of hemorrhage, necrosis, or its age.

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These tumors may be firm or gelatinous. Considerable regional variation in appear-ance is characteristic. Secondary to necrosis, some areas are firm and white, some are soft and yellow, and still other are cystic with local hemorrhage. Moreover, glioblastomas have a large variability in size and infiltration beyond the visible tu-mor margin is always present [62].

Glioblastoma forms in the cerebral white matter and grows quickly along the fibers or along vessels and it seems that it follows the physical structures in the extracel-lular matrix of the neighboring brain [23]. The tumor may arise in any part of the Central Neural System but the frontal and the temporal lobes are the most fre-quently affected. The central part of the brain is often involved, and a classical appearance is provided by the butterfly type, in which the tumor extends in both hemispheres [27] as depicted in figures 1.4. Malignant cells carried in the cere-brospinal fluid may invade and migrate away from the main tumor within the brain; however, glioblastomas rarely spread elsewhere in the body.

(a) C+ T1 WI Coronal (b) C+ T1 WI Axial

Figure 1.4: C+ (contrast-enhanced) T1 MRI WI (weighted) of a butterfly glioblas-toma. If the tumor arises in the central part of the brain, it spreads in both hemi-spheres and thus it shows a butterfly appearance.

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1.3.2 Diagnosis and treatment

Brain tumors are very difficult to be treated due to the following factors: tu-mor cells are very resistant to conventional therapies, many drugs cannot cross the blood brain barrier to act on the tumor and brain has a very limited capacity to repair itself [34]. Moreover, concerning glioblastoma, its treatment remains even more difficult. As a matter of fact, this tumor is heterogeneous, it is often lo-cated in a region that is beyond the reach of local control and its highly diffusive nature makes the complete removal almost impossible. Therefore, the treatment of patients with malignant gliomas is made on different levels and encompasses surgery, radiotherapy, and chemotherapy.

For the diagnosis, conventional imaging techniques, such as computed tomog-raphy (CT or CAT scan) and magnetic resonance imaging (MRI), are commonly used to pinpoint the tumor. Depending on the patient, other exams could be re-quested; e.g. magnetic resonance spectroscopy (MRS) is used to examine the tu-mor’s chemical profile and determine the nature of the lesions seen on the MRI. Upon an initial diagnosis, the first step in treating the glioblastoma is to remove as much of tumor as possible, without injuring brain tissue (figure 1.5). In the last years, the diffusion tensor imaging technique, which will be described in more details in Chapter 2, has gained in importance since it provides parametric maps that help to visualize different aspects of the tissue micro-structure which may help the neurosurgeons to avoid disrupting important nerve connections within the brain itself. However, high-grade tumors are surrounded by a zone of migrating, infiltrat-ing tumor cells that invade surroundinfiltrat-ing tissues and thus removinfiltrat-ing the entire tumor could be very difficult and, almost always, impossible. Anyway, gross tumor re-section immediately decompresses the brain reducing the intracranial pressure and, due to the consequent reduction in neoplastic cells in the surgical cavity, probably increases the likelihood of response to radiotherapy and/or chemotherapy; it may, moreover, delay progression. Always after surgery, when the wound is healed,

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radiation therapy can begin. The goal of radiation therapy is to kill tumor cells selectively while leaving normal brain tissue unharmed. In standard external beam radiation therapy, multiple sessions of standard-dose ”fractions” of radiation are delivered to the tumor site as well as a margin in order to treat the zone of infiltrat-ing tumor cells. Each treatment induces damage to both healthy and normal tissue but by the time the next treatment is given, most of the normal cells have repaired the damage, but the tumor tissue has not. In addition, the patient is treated con-comitant with chemotherapy, by which special drugs designed to kill tumor cells are administrated. Nowadays, the standard drug used is the temozolomide, which has shown to give survival benefits with minimal additional toxicity. The current standard of care, that is the Stupp protocol, has led to a significant improvement in patient survival. This protocol consists of adjuvant radiotherapy after resection plus continuous daily temozolomide (75 mg per square meter of body-surface area per day, 7 days per week from the first to the last day of radiotherapy), followed by six cycles of adjuvant temozolomide (150200 mg per square meter for 5 days during each 28-day cycle) [68]. Because traditional methods of treatment are un-likely to result in a prolonged remission of GBM tumors, researchers presently are investigating several innovative treatments in clinical trials [49, 74].

(a) T1 Axial (b) Post T1 MDC- Axial

Figure 1.5: Example of glioblastoma resection. The figures have been gently pro-vided by the Istituto Neurologico Carlo Besta.

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Diffusion Tensor Imaging

Diffusion imaging is a magnetic resonance imaging method introduced in the mid-1980s [37]. It focuses on the random translational motion of molecules, mainly water, which, during the typical diffusion time of about 50 msec [37, 47], move in the brain over a distance of 10µm [37, 47], bouncing, crossing, or interacting with many tissue components such as cell membrane, fibers and macromolecules. Since these movements encounter different obstacles and vary in according to the tissue or certain pathological modifications (such as intracellular edema, abscess, tumor), diffusion data provides indirect information on the structure surrounding these wa-ter molecules and on the geometric organization of the brain. Indeed, the overall effect observed in a diffusion MRI image voxel reflects, on a statistical basis, the displacement distribution of the water molecules present within this voxel and it al-lows to probe tissue structure at a microscopic scale, well beyond the usual image resolution. A useful analogy is the shape of ink dropped on a piece of paper. After the ink is dropped, it begins to spread due to the thermal motion of its molecules and the shape of the ink stain can reveal something about the underlying structure of the paper. If the shape of the ink stain is circular, it is called isotropic diffusion. If the stain is elongated along a precise direction, it is called anisotropic diffusion. Diffusion-weighted (DW) magnetic resonance (MR) imaging is currently the only MR imaging technique that provides in vivo information on water diffusion within

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tissues, involving the use of phase-defocusing and phase-refocusing gradients. We briefly describe this technique in the first section of the present chapter.

A further development of DWI is the diffusion tensor imaging (DTI) technique, which was introduced in the mid-1990s [37]. It studies the directions of wa-ter molecule motion to characwa-terize diffusion behavior in biological tissues and it makes use of the concept of anisotropy to estimate the axonal organization of the brain. Namely, water moves more easily along the axonal bundles rather than perpendicular to those bundles for the fact that there are fewer obstacles in that di-rection. Thus, studying the directions of diffusion of water molecules, it is possible to gain access to the microscopic organization of the tissue in which the diffusion takes place. The information of the anisotropic structure of the tissue are summa-rized in a tensor of diffusivity, which gives the name to this technique. We describe DTI method in section 2.

2.1 Diffusion Weighted Measurements

In order to better understand the diffusion tensor imaging technique, it is useful to briefly illustrate how diffusion-weighted imaging (DWI) works. In particular, in the following we explain how the MR signal is recorded and the relationship between the diffusion of water molecules in tissues and the intensity of the sig-nal measured by MR imaging. As mentioned, diffusion weighted imaging (DWI) focuses on measuring the random Brownian motion of water molecules within a voxel of tissue and it is based on diffusion weighted sequences by which medical images are obtained. These images present a contrast which is namely influenced by the difference of motion of water molecules.

The information that can be acquired from a DW MRI image are: S, the MR signal recorded into a voxel, P D, the proton density which represents the water concen-tration, T1 and T2, the relaxation times (decay times) after excitation which are

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are related by the equation S = P D 1 − e−T R/T1 e−T E/T2_e−bD _(2.1)

where T R and T E are respectively related to the timing of excitation (called repe-tition time) and the preparation period (called echo time) of the MR signal whereas b is the diffusion weighted factor. In this equation it is important to note that the magnitude of signal from water S is the information we obtain from the MR scan-ners while T R, T E and b are imaging parameters that we can control and, changing these parameters, we can change the contribution (weighting) of T1,T2 and D to

the signal [47].

To understand the b term, we first have to introduce the concept of magnetic field gradient pulse. In figure 2.1, the effect of a gradient pulse is explained with a schematic diagram.

We consider the signals generated by two different water molecules (red and blue circles) in two different position along the Y axis (for instance). In time period I, both molecules see the homogeneous magnetic field applied and thus the spins have the same frequency. Then, in time period II, the Y gradient is applied and the spin in blue position is exposed to a lower magnetic field than the one in the red po-sition and so it resonates at a lower frequency. After the gradient pulse ends (time period III), the signals from the water molecules begin to have the same frequency but their phases are shifted according to

∆ϕ = ϕ1− ϕ2 = γGδ(y1− y2) (2.2)

where γ is the gyromagnetic ratio of proton, G is the constant gradient strength, δ is the duration of the gradient pulse application and yi the position of the spin

with respect to the gradient. In time period IV, an other Y gradient is applied with the opposite polarity and, interestingly, the phase shift is unwound. In fact, in this period, water molecules in the blue position start to resonate at higher frequency

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Figure 2.1: Magnetic field gradient and their effects on the MRI signal [47]. Y gradients are applied at time periods II and IV. The durations of the period II and IV are the same, but the orientations of the gradient are opposite. Water signals from two different locations are shown by red and blue colors. The signal frequency is proportional to the strength of the magnetic field B0, represented by the green

arrows. The figure is taken in [47].

and then the perfect refocusing of the phase occurs.

However, the perfect refocusing of the phase occurs only if the particles do not change their initial position. For this reason, in the diffusion measurement, the phase difference ∆ϕ is used to compute water motion in the direction of the gradi-ent of the magnetic field.

In order to better explain the concept, we consider the figure 2.2, in which water molecules and the relative spin phase, sketched as circles with black arrows, are illustrated within a voxel. When the first gradient is applied, a phase shift depend-ing on the position of the spin is introduced between the molecules and the spins are dephased. After a time interval ∆t, a second gradient pulse for the phase refo-cusing is applied. The reforefo-cusing is perfect only when the water molecules do not move between the two pulses. If there is translational motion of water molecules, perfect refocusing would fail. Because the signal at each voxel represents the sum

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of the signal from all the water molecules in that sample, the imperfect refocusing leads to a signal loss. It is important to notice that, as show by equation (2.1), the higher the diffusion coefficient D is, the more signal loss we expect. The term b is related to the intensity of the gradient application G and to the time separation between the two gradient pulses ∆t [44] :

b = γ2G2δ2 ∆t −δ 3 . (2.3)

The important point is that we can control the amount of the b values as all the quantities in (2.3) can be set, and we can expect a different amount of signal loss. In fact, the longer the separation ∆t is and the more time there is for water molecules to move around which is reflected in a higher signal loss.

Figure 2.2: The effect of gradients application on the motion of water molecules. If there is molecular diffusion, a total phase shift is introduced. In absence of water motion, gradients have no consequence. The figure belongs to [47].

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Consequently, in order to compute the diffusion coefficient of water molecules it is sufficient to perform two measurements with different b values, while other imaging parameters (T E and T R) remain the same. In particular, one of the two measurements must be done with a null b value. So, the signal loss due to the diffusion precess is governed by the Stejskal-Tanner equation [67]:

S = S0e−bD (2.4)

where S0 is the signal intensity without diffusion sensitization, that is obtained

from (2.1) setting b = 0. In this way, the diffusion coefficient, D, can be calculated from the signal intensity differences between these two studies:

D = −1 bln S S0 . (2.5)

Since biological tissues can be considered anisotropic mediums because of the architecture of the tissue itself or the presence of obstacles, the D diffusion param-eter is substitute with an apparent diffusion coefficient ADC [36]:

S = S0e−bADC. (2.6)

The reason for which the adjective apparent is used is related to the fact that what is measured is not the real diffusion coefficient. For instance, the ADC of water in parenchymia is much smaller than that of the cerebrospinal fluid. This could be due to the more viscous environment but also due to many obstacles. Namely, when the barriers are aligned along one orientation, the ADC depends on the mea-surement orientation, e.g. meamea-surements along the structures lead to higher ADC (less obstacles) and viceversa measurements perpendicular to them lead to smaller ADC (more barriers).

In other words, to sum up the present section, in order to estimate the apparent dif-fusion coefficient along a specific direction it is necessary to apply a magnetic field

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Figure 2.3: In practice, different degrees of diffusion-weighted images can be ob-tained using different values for the b factor. The larger the b factor, the more the signal intensity (S) becomes attenuated in the image. This attenuation is modu-lated by the diffusion coefficient. Infact, the signal in structures with fast diffusion (for example, water-filled ventricular cavities) decays rapidly with increasing b, whereas the signal in tissues with low diffusion (for example, grey and white mat-ter) decreases more slowly. By fitting the signal decay as a function of b, one ob-tains the apparent diffusion coefficient (ADC) for each elementary volume (voxel) of the image, [35]. Nowadays, it is used a b value in the range of 700-1000 s/mm2, [78]

along that direction and perform two measurements, one of which with a null b: thus, a misure along the X, Y , and Z axis will lead to the computation of ACDx,

ACDy and ACDz. In the next section, the diffusion tensor imaging, a technique

that summarize and the information from the ACDs is introduced.

2.2 Diffusion Tensor Imaging Technique

In tissues, such as brain gray matter, where the measured apparent diffusivity is largely independent of the orientation of the tissue (i.e. isotropic), it is usu-ally sufficient to characterize the diffusion proprieties of the tissue with the ADC. However, in anisotropic media, such as in white matter, where the measured diffu-sivity is known to depend upon the orientation of the tissue, a single ADC cannot characterize the orientation-dependent water mobility. In this case, a mathemat-ical object, describing molecular mobility along each direction and correlations

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between these directions, is required. This object is a second order tensor called diffusion tensor[3] and it consists of 9 components that are included in a 3 × 3 symmetric matrix (i.e. Dij = Dji, with i, j = x, y, z). The diffusion tensor D has

the following form:

D =      Dxx Dxy Dxz Dxy Dyy Dyz Dzx Dzy Dzz     

where Dxx, Dyy, Dzzrepresent the magnitude of diffusivity along the x, y, z

direc-tions and Dxy, Dxz, Dyzrepresent the magnitude of diffusion along one direction

arising from a concentration gradient in an orthogonal direction (see figure 2.4). According to (2.4), the signal attenuation of the MR signal is

S = S0e−bD = S0e− P ijbijDij _(2.7) = S0e−(bxxDxx+byyDyy+bzzDzz+2bxyDxy+2bxzDxz+2byzDyz) (2.8) with bij = γ2GiGjδ2 ∆t − δ 3 . (2.9)

In this formalism, the diffusion-weighting factor b incorporates the direction and magnitude of the applied diffusion gradient vector (Gx, Gy, Gz).

In order to describe the diffusion, the components of the tensor have to be evaluated. Since the tensor is symmetric, only six components are independent and consequently seven diffusion measurements are required at least for each voxel of the image: one measurement is necessary to evaluate S0, while six measurements

are made along six non collinear directions.

We remark that the values of the components depend on the orientation of the sample respect to the laboratory reference system, defined by the directions of the spatial encoding gradients. Hence, the tensor cannot be uniquely described. However, it exists a reference frame [x0, y0, z0] that coincides with the principal directions of diffusivity and thus it is determined by the local anatomy. In the

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Figure 2.4: Components of the diffusion tensor D. Diagonal elements Dxx, Dyy,

Dzz are proportional to the diffusion displacement coefficients (ADCs) along the

three directions of the experiment coordinate system. Off-diagonal elements are proportional to the correlations (covariances) of displacements along these direc-tions (figure from [24]).

reference frame, the off diagonal terms are null (i.e. the orthogonal directions appear non correlated) and the tensor is reduced to a diagonal matrix (figure 2.5). This is called main axes reference system and it can be mathematically obtained by the diagonalization of the diffusion tensor. Hence, the problem consists in finding the eigenvalues and the eigenvectors of the matrix D:

Dei = λiei for i = 1, 2, 3 (2.10)

which in matrix form is

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Figure 2.5: Diffusion components in relation to fibers orientation. Figure (a): Dxx

and Dyyare determined by measurements along the x− and y− directions. Figure

(b): the values are computed in the reference system which coincides with the direction of the fibers. The figure is taken from [24].

with E = ( e1| e2| e3) and Λ =      λ1 0 0 0 λ2 0 0 0 λ3      (2.12)

where Λ is the matrix of the eigenvalues and E is the orthogonal matrix of the eigenvectors.

The most intuitive way to conceptualize the information provided by the diffu-sion tensor is to view it geometrically by the diffudiffu-sion ellipsoid [3]. An ellipsoid is a 3D representation of the diffusion distance covered in space by molecules in the diffusion time interval ∆. This ellipsoid, which can be displayed for each voxel of the image, is easily calculated from the eigen-diffusivities. Its equation is

e1 √ 2λ1∆ 2 + e2 √ 2λ2∆ 2 + e3 √ 2λ3∆ 2 = 1. (2.13)

In particular, the three eigenvectors e1, e2, e3 and the three eigenvalues λ1, λ2, λ3

describe respectively the directions and lengths of the ellipsoid’s axes in descend-ing order of magnitude (see figure 2.7, first row). The largest eigenvector, termed the ”primary eigenvector ” and its associated eigenvalue λ1 indicate, respectively,

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eigenvec-tor is important for fiber tractography algorithms because this veceigenvec-tor indicates the orientation of axonal fiber bundles. Therefore, λ1is also termed longitudinal

dif-fusivity, because it specifies the rate of diffusion along the orientation of the fibers. The second and third eigenvectors are orthogonal to the primary eigenvector, and their associated eigenvalues λ2and λ3give the magnitude of diffusion in the plane

transverse to axonal bundles. Hence, they are also known as radial diffusivities. The figure 2.6 shows a schematic depiction of diffusion properties through the dif-fusion ellipsoid scheme.

Finally, it is possible to analyzed the diffusion tensor data by a number of rota-tionally invariant diffusion parameters, derived from the 3 eigenvalues, in order to provide information on tissue microstructure and architecture for each voxel. The mean diffusivity Dav, which characterizes the overall mean-squared

displace-ment of molecules and the overall presence of obstacles to diffusion within a voxel (average ellipsoid size), is evaluated as follows:

Dav =

T r(D)

3 =

Dxx+ Dyy+ Dzz

3 . (2.14)

Among the various scalar indexes proposed to characterize diffusion anisotropy, the most widely used is the fractional anisotropy (FA) which describes how much molecular displacement vary in space (ellipsoid eccentricity) and it is related to the presence of oriented structures. Its formula is:

F A = r 1 2 p((λ1− λ2)2+ (λ2− λ3)2+ (λ3− λ1)2) pλ2 1+ λ22+ λ23 . (2.15)

This is a very convenient index because it is scaled from 0, isotropic medium, to 1, anisotropic medium, indicating perfectly linear diffusion occurring only along the primary eigenvector. The fiber orientation information inherent in the primary eigenvector can be visualized on 2D images by assigning a color to each of 3 mu-tually orthogonal axes (see figure 2.7, bottom right).

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Figure 2.6: Scheme of diffusion properties via ellipsoid visualization for isotropic unrestricted diffusion, isotropic restricted diffusion, and anisotropic restricted dif-fusion [48].

Figure 2.7: Depiction of the eigenvalues, mean diffusivity, FA and a color map of the fiber orientation. In the first row, λ1, λ2, λ3 are shown with the same intensity

scaling. The eigenvalues are always ordered in descending order of intensity with the first eigenvalue being the greatest. In the second row, the mean diffusivity, the fractional anisotropy and a color map of the orientation of the primary eigenvector are shown. The figure is taken from [48].

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Mathematical Model

In this chapter, we provide the derivation of a mathematical model suitable to describe a tumoral mass growth. The theoretical framework of our approach comes from the continuum theory of mixture, which is briefly introduced in section 3.1. The theory of mixture, which has its roots in the continuum mechanics, is based on balance equations and conservation principles and it has been widely applied to systems which can be studied as a mixture of interacting continua. In the second section, we present a two-species diffuse interface model of tumor growth. It is a general model formulated for a tumor comprising viable tumor cells and water, i.e. interstitial fluid which may contain dead cells and other substances. A single vital nutrient (e.g. oxygen) is taken into account in the model and it is assumed that the density of the extra-cellular matrix (ECM) remains constant in time and space and does not significantly degrade or remodel as the tumor mass grows. In the last part of the chapter, we specifically deal with glioblastoma and we estimate the biological parameter introduced in our mathematical model.

3.1 A continuum theory of mixtures

The starting point of our research is the theory of mixture. The literature on this subject is very rich, dating back to the early work on simple mixtures of Fick

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and Darcy during the middle of the XIX century and progressing to the general continuum theories advanced by Truesdell, the pioneer of the modern continuum models of mixture fluids, in the last years of the 50s [51]. The purely mechanical model developed by Truesdell was essentially extended and corrected by Muller (1968), who gave rise to the modern thermodynamical theory of miscible mixtures [82]. Then, the important and comprehensive article of Bowen [6] and the mono-graph of Rajagopal and Tao [63] provided a wide review of the relevant literature on the subject.

Inspired by the article of Bowen [6], in the following we report some basic notations of the mixture theory and we provide a brief discussion on the mass balance equation under the incompressibility condition.

3.1.1 Kinematics

The fundamental idea underlying the mixture theory is that a material body or a mixture B can be composed of N constituent species called phases B1, B2, ..., BN

and in any instant of time, each point of the domain is occupied simultaneously by particles of all components. It is convenient to describe B in the Eulerian reference and represent all fields of the model as functions of the spatial variable x ∈ B and time t ∈ T . We introduce a fixed reference configuration so that the spatial position occupied by a material point at time t is

x = χα(Xα, t) (3.1)

where χαis the deformation function of the αthconstituent and Xαis the position

of a particle of the αthconstituent in its reference configuration. Then, the velocity of a particle in Xαat time t is defined by

vα=

∂χα(Xα, t)

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Thus, if Ψ is any function of (x, t), the material derivative of Ψ following the motion of the αthconstituent is

˙ Ψα=

∂Ψ(x, t)

∂t + ∇Ψ(x, t)vα. (3.3)

Given V the total volume of B containing a point (x, t) and mα(x, t) the mass of a

single constituent Bα, we define the apparent mass density ραof each constituent

αthas

ρα(x, t) =

mα(x, t)

V (x, t) (3.4)

and consequently the apparent mass density of the mixture at (x, t) is

ρ(x, t) =

N

X

α=1

ρα(x, t). (3.5)

Then, we introduce the true mass density γαof the αthconstituent by the relation

γα(x, t) =

mα(x, t)

Vα(x, t)

(3.6)

where Vα(x, t) is the volume occupied by a single Bα. It is then possible to define

a new quantity φα, as follows:

φα(x, t) =

ρα(x, t)

γα(x, t)

= Vα(x, t)

V (x, t). (3.7)

This quantity is called volume fraction and it physically represents the volume of the αth constituent per unit volume of the mixture. Consequently, we have the following relation for each αthconstituent:

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A mixture is said to be saturated if the constituents satisfy

N

X

α=1

φα(x, t) = 1. (3.9)

Finally, it is useful to introduce the concept of incompressiblility. In a theory of mixtures, the αth constituent is incompressible if ˙γα is zero which physically

means that γα can only depend on Xα. Consequently, it is usually assumed that,

when the αthconstituent is incompressible, γαis constant. In any case, the mixture

is incompressible when every constituent in the mixture is incompressible.

3.1.2 The mass balance equation for mixtures

We suppose that the general mixture B occupies a fixed spatial region Ω in R3 with surface ∂Ω and we introduce a general term modeling the inter-component mass exchange per unit volume Γα(x, t). Each of the N constituents must satisfy

its own mass balance equation, which written in the integral formulation reads ∂ ∂t Z Ω ραdV = − I ∂Ω ραvα· ndS + Z Ω ΓαdV, (3.10)

where n is the outward vector of the element of surface dS. Requiring that (3.10) holds for every spatial volume, we can localize the mass balance equation for the αth_{constituent and, by means of the divergence theorem, we can obtain the}

Eule-rian local form of (3.10) which is ∂ρα

∂t + ∇ · (ραvα) = Γα. (3.11) Then using equation (3.3) and the vector calculus identity ∇ · (f ψ) = ψ · ∇f + f ∇ · ψ, where f is a scalar function and ψ a vector field, equation (3.11) yields to

˙